Show All Work To Multiply $(2+\sqrt{-25})(4-\sqrt{-100})$.

Introduction

In this article, we will delve into the world of complex numbers and explore the process of multiplying them. Complex numbers are a fundamental concept in mathematics, and understanding how to multiply them is crucial for solving various mathematical problems. We will use the given expression $(2+\sqrt{-25})(4-\sqrt{-100})$ as an example to demonstrate the step-by-step process of multiplying complex numbers.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2=-1$. The real part of a complex number is denoted by $a$, and the imaginary part is denoted by $b$. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Multiplying Complex Numbers

To multiply complex numbers, we can use the distributive property, which states that for any complex numbers $a+bi$ and $c+di$, the product is given by:

$(a+bi)(c+di) = ac + adi + bci - bd$

This can be rewritten as:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Step-by-Step Solution

Now, let's apply this formula to the given expression $(2+\sqrt{-25})(4-\sqrt{-100})$. We can start by simplifying the square roots:

$\sqrt{-25} = \sqrt{-1}\sqrt{25} = 5i$

$\sqrt{-100} = \sqrt{-1}\sqrt{100} = 10i$

Now, we can substitute these values into the original expression:

$(2+5i)(4-10i)$

Using the distributive property, we can multiply the two complex numbers:

$(2+5i)(4-10i) = (2\cdot4 - 5\cdot10i) + (2\cdot(-10i) + 5\cdot4)i$

Simplifying the expression, we get:

$(2+5i)(4-10i) = (8 - 50i) + (-20i + 20)i$

Combining like terms, we get:

$(2+5i)(4-10i) = (8 - 50i - 20i + 20i^2)$

Since $i^2=-1$, we can substitute this value into the expression:

$(2+5i)(4-10i) = (8 - 50i - 20(-1))$

Simplifying further, we get:

$(2+5i)(4-10i) = (8 - 50i + 20)$

Combining like terms, we get:

$(2+5i)(4-10i) = (28 - 50i)$

Conclusion

In this article, we have demonstrated the step-by-step process of multiplying complex numbers using the distributive property. We have used the given expression $(2+\sqrt{-25})(4-\sqrt{-100})$ as an example to illustrate the process. By following these steps, we can multiply complex numbers and simplify the resulting expression.

Common Mistakes to Avoid

When multiplying complex numbers, it's essential to remember the following common mistakes to avoid:

• Not simplifying the square roots: Make sure to simplify the square roots before multiplying the complex numbers.
• Not using the distributive property: Use the distributive property to multiply the complex numbers.
• Not combining like terms: Combine like terms to simplify the resulting expression.

Real-World Applications

Multiplying complex numbers has numerous real-world applications, including:

• Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
• Navigation: Complex numbers are used to represent navigation systems and analyze their performance.

Final Thoughts

Introduction

In our previous article, we explored the process of multiplying complex numbers using the distributive property. In this article, we will answer some frequently asked questions about multiplying complex numbers.

Q: What is the difference between multiplying complex numbers and multiplying real numbers?

A: The main difference between multiplying complex numbers and multiplying real numbers is that complex numbers have an imaginary part, which is represented by the letter $i$. When multiplying complex numbers, we need to use the distributive property and combine like terms to simplify the resulting expression.

Q: How do I simplify the square roots of complex numbers?

A: To simplify the square roots of complex numbers, you can use the following formula:

$\sqrt{-a} = i\sqrt{a}$

For example, $\sqrt{-25} = i\sqrt{25} = 5i$.

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

This formula can be rewritten as:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Q: How do I multiply complex numbers with different powers of $i$?

A: When multiplying complex numbers with different powers of $i$, you can use the following rules:

• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$

For example, $(2+3i)(4-5i) = (2\cdot4 - 3\cdot5i) + (2\cdot(-5i) + 3\cdot4)i$

Q: Can I multiply complex numbers with negative coefficients?

A: Yes, you can multiply complex numbers with negative coefficients. When multiplying complex numbers with negative coefficients, you can use the following rules:

• $-a = -1 \cdot a$
• $-bi = -1 \cdot bi$

For example, $(-2+3i)(4-5i) = (-2\cdot4 - 3\cdot5i) + (-2\cdot(-5i) + 3\cdot4)i$

Q: How do I multiply complex numbers with fractions?

A: When multiplying complex numbers with fractions, you can use the following rules:

• $\frac{a}{b} = a \cdot \frac{1}{b}$
• $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

For example, $\frac{2}{3} + 3i \cdot \frac{4}{5} = \frac{2}{3} \cdot \frac{4}{5} + 3i \cdot \frac{4}{5}$

Q: Can I multiply complex numbers with imaginary numbers?

A: Yes, you can multiply complex numbers with imaginary numbers. When multiplying complex numbers with imaginary numbers, you can use the following rules:

• $ai \cdot bi = -ab$
• $ai \cdot bi = -ab$

For example, $(2+3i)(4+5i) = (2\cdot4 - 3\cdot5i) + (2\cdot5i + 3\cdot4)i$

Conclusion

In this article, we have answered some frequently asked questions about multiplying complex numbers. We have covered topics such as simplifying square roots, multiplying complex numbers with different powers of $i$, and multiplying complex numbers with negative coefficients. By following these rules and using the distributive property, you can multiply complex numbers and simplify the resulting expression.

Common Mistakes to Avoid

When multiplying complex numbers, it's essential to remember the following common mistakes to avoid:

• Not simplifying the square roots: Make sure to simplify the square roots before multiplying the complex numbers.
• Not using the distributive property: Use the distributive property to multiply the complex numbers.
• Not combining like terms: Combine like terms to simplify the resulting expression.

Real-World Applications

Multiplying complex numbers has numerous real-world applications, including:

• Electrical Engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
• Signal Processing: Complex numbers are used to represent signals and analyze their frequency content.
• Navigation: Complex numbers are used to represent navigation systems and analyze their performance.

Final Thoughts

In conclusion, multiplying complex numbers is a fundamental concept in mathematics that has numerous real-world applications. By following the rules and using the distributive property, you can multiply complex numbers and simplify the resulting expression. Remember to avoid common mistakes and use the distributive property to multiply complex numbers. With practice and patience, you'll become proficient in multiplying complex numbers and apply this skill to real-world problems.